4.4 4.5M6综合卷
浙教版-数学六年级下册第四、五单元综合测试卷

浙教版数学六年级下册第四、五单元综合测试卷姓名:________ 班级:________ 成绩:________小朋友,带上你一段时间的学习成果,一起来做个自我检测吧,相信你一定是最棒的!一、选择题1 . △代表一个不为0的自然数.那么,得数最大的是()A.△×B.△÷C.÷△2 . 一盒有净含量为750毫升的长方体盒装酸奶,量得外包装长8厘米,宽5厘米,高15厘米,根据以上数据,你认为净含量的标准是()A.真实B.虚假C.无法确定3 . 某教育局装备科购进96台电脑,按4∶5∶3分发给第一、第二和第三小学,三所小学各发到电脑多少台?正确的解答是()A.第一小学:22台,第二小学:45台,第三小学:29台B.第一小学:32台,第二小学:40台,第三小学:24台C.第一小学:30台,第二小学:50台,第三小学:16台D.第一小学:20台,第二小学:60台,第三小学:20台4 . 某校三个班学生共有157人,且三个班的男生人数都相等.第一班男生占全班的,第二班男生占全班的,那么第三班的女生有()人.A.21B.28C.35D.425 . 和奇数K相邻的两个奇数是()A.K-1 和K+1B.K-1和K+ 3C.K-2和K+ 2D.K-3和K+ 36 . a和b的最大公因数只有1,则a和b的最小公倍数是().A.a B.b C.ab D.17 . 一件商品先涨价10%,后降价10%,这件商品最后价格()A.比原价降了1%B.不变C.比原价涨了1%D.比原价降了2%8 . 甲、乙两地相距4.5千米,画在一幅图上长1.5厘米,这幅图的比例尺是()。
A.1:3B.3:1C.1:3000009 . 比a的5倍多15的数是()A.5a-15B.5a+15C.15-5a D.(a+15)×510 . 小明把6X-8错写成6(X-8),结果比原来()A.多8B.少8C.少40D.多32二、填空题11 . 50 6008 6540读作,改写成用“万”作单位是,省略亿位后面的尾数约是亿.12 . 等腰三角形的一个底角是35°,这个三角形的顶角是(__________)。
外研版六年级英语上册Modules 4~5综合达标卷含答案

外研版六年级英语上册Modules 4~5综合达标卷时间:40分钟满分:100分听力部分(30分) 笔试(70 分)总分题号一二三四五六七八九十十一十二得分第一部分听力(30 分)一、听录音, 根据所听内容选出你所听到的内容。
(5 分)()1. A. fly kites B. sing songs C. write emails()2. A. meet B. met C. meal()3. A. speak English B. write in English C. English stamps ()4. A. dance B. friends C. address()5. A. sounds nice B. looks nice C. sounds good 二、听录音, 根据所听内容给下列图片排序。
(5 分)() () () () ()三、听录音, 填写单词补全句子。
(10 分)1. At the Dragon Boat Festival, we all go to see the dragon boat ____________.2. Thanksgiving is my ____________ festival.3. We fly the ____________ and we ____________ songs.4. I can write ____________ in English.5. We always have a ____________ meal at the Spring Festival.四、听录音, 根据所听内容选择正确的答案。
(10 分)()1. Where does Simon live?A. China.B. New York.()2. What does Simon like?A. He likes flying a kite in the park.B. He likes riding a bicycle and reading books.()3. Who is Simon’s friend?A. Laura.B. Mary.()4. Laura can write letters in ________.A. EnglishB. Chinese()5. Mary can write letters to Laura in ________.A. EnglishB. Chinese第二部分笔试(70 分)五、选出与所给单词同类的一项。
牡丹江2024年07版小学六年级第4次英语第6单元综合卷

牡丹江2024年07版小学六年级英语第6单元综合卷考试时间:80分钟(总分:110)考试人:_________题号一二三总分得分评级介绍:本试卷针对六年级学生,涵盖第6单元的核心知识点,难度适中,重点考察词汇运用和句型理解,旨在提升学生的综合英语能力。
一、(选择题)总分:45分(1分/题)1、What do you call the time when everyone eats together?中文解释:大家一起吃饭的时间叫什么?A, Meal timeB, Snack timeC, Break time2、How many teeth does an adult human normally have?A, 20B, 24C, 28D, 323、What is the English word for "植物经济效益分析"?A, Plant economic benefit analysisB, Agricultural analysisC, Environmental analysisD, Ecological analysis4、Which type of plant is a cucumber?中文解释:黄瓜属于哪种植物?A, FruitB, VegetableC, Herb5、What is a popular drink made with tea leaves?A, JuiceB, CoffeeC, TeaD, Soda6、What is the English translation of "火烈鸟"?A, FlamingoB, PelicanC, HeronD, Crane7、What is the English word for "游戏"?A, SportB, GameC, PlayD, Contest8、What does "植物区域发展" mean in English?A, Plant regional developmentB, Agricultural developmentC, Environmental developmentD, Ecological development9、What is the color of a snowflake?中文解释:雪花是什么颜色的?A, WhiteB, BlueC, Gray10、Where do birds live?A, In the waterB, In the skyC, In treesD, On the ground11、What is the name of the fruit that is green on the outside and pink on the inside?A, KiwiB, WatermelonC, Grapefruit12、What is the English translation of "大鲨鱼"?A, Great white sharkB, Hammerhead sharkC, Tiger sharkD, Bull shark13、选出与所给单词同类的一项。
锦州2024年05版小学4年级第六次英语第4单元综合卷

锦州2024年05版小学4年级英语第4单元综合卷考试时间:80分钟(总分:120)A卷考试人:_________题号一二三四五总分得分一、综合题(共计100题)1、What is the name of the famous dog in the cartoon "Snoopy"?A. CharlieB. WoodstockC. SnoopyD. Garfield答案:C2、In which country is the Eiffel Tower located?A. ItalyB. GermanyC. FranceD. Spain3、填空题:My mom loves taking pictures of ____.4、填空题:My friend has a unique __________ (风格).5、填空题:The goldfish can be many different _______ (品种).6、What do we call the process of converting agricultural waste into useful products?A. Biomass conversionB. CompostingC. Anaerobic digestionD. All of the above答案: D. All of the above7、填空题:I saw a _______ (小兔子) hopping in the grass.8、听力题:She is a good ________.9、填空题:The ______ (触感) of leaves can vary widely.10、听力题:Many _______ can have multiple uses.11、选择题:What is the name of the famous character who goes on adventures in Neverland?A. Peter PanB. WendyC. Tinker BellD. Captain Hook12、填空题:We need to water the ______ every day.13、What is the opposite of "happy"?A. SadB. ExcitedC. AngryD. Tired答案: A14、填空题:I love my new ________ that's shaped like a dinosaur.15、听力题:Rainforests are known for their rich ______.16、填空题:A ________ (植物观察志愿者) contributes to knowledge.17、听力题:Gravity makes things fall _______.18、填空题:I want to learn about different _______ (文化). It broadens my _______ (视野).19、听力题:The bear is searching for berries in the ____.20、What is the main ingredient in pizza dough?A. RiceB. FlourC. CornD. Oats答案: B21、听力题:A chemical formula shows the elements present in a ______.22、填空题:My favorite animal is the ______. I think it is very ______ and interesting. This animal is known for its ______ fur and big ______. It lives in ______ and often eats ______.23、What do we call the process by which a caterpillar becomes a butterfly?A. MetamorphosisB. EvolutionC. TransformationD. Development答案:A24、选择题:What is the color of a typical fire?A. BlueB. RedC. YellowD. Both B and C25、听力题:The ______ moves slowly and has a shell.26、听力题:The element with atomic number is __________.27、Listen and number.(听音标号。
冀教版数学六年级下册第四、五单元综合素质达标卷(含答案)

第四、五单元综合素质达标一、填空。
(每空 1 分,共 23 分)1.7.5m 3=( )dm 3 108dm 2=⎝ ⎛⎭⎪⎫ m 2 5.18L =( )L( )mL 8m360dm 3=( )m 32. 把一根长是 4m ,底面半径是 2dm 的圆柱形木料截成 4 段相同的小圆柱,表面积比原来增加了( )dm 2。
3. 右图是一个( ),3dm 是它的( ),10dm 是它的( ),它的体积是( )dm 3,与它等底等高的圆柱的体积是( )dm 3。
4. 一个圆柱形木料,底面半径是 3cm ,高是 8cm ,这个圆柱的表面积是 ( ) cm 2。
如果把它削成一个最大的圆锥,削去部分的体积是( )cm 3。
5.从顶点沿着圆锥的高把一个圆锥切成完全相同的两部分, 切面是一个底边长4cm 、高 6cm 的等腰三角形,原来这个圆锥的体积是( )cm 3。
6. 梦梦把右图这块长方体橡皮泥捏成一个高是 8cm 的圆柱,捏成的圆柱的底面积是( )cm 2。
如果捏成与圆柱底面相等的圆锥,这个圆锥的高是( )cm。
(橡皮泥无损耗)7.如图,这根圆木的底面直径是( )m。
如果将这根圆木加工成最大的方木,方木的体积是( )m3,加工成的方木与原来圆木的体积比是( )。
8.一个圆锥和一个圆柱的体积相等,圆锥和圆柱底面半径的比是3:2,圆锥的高是16cm,圆柱的高是( )cm。
9.******************是某位老师的身份证号码,这是一位( )老师(填“男”或“女”) ,2024 年( )岁。
10.如果密码锁的密码是由三个数字组成的,那么0 ~9 这十个数字能组成( )个数字不重复的密码。
二、选择。
(将正确答案的字母填在括号里。
每题 2 分,共16 分)1.营养学家建议,儿童每天水的摄入量约为1500mL,要达到这个要求,明明每天用底面内直径为8cm、高为10cm 的圆柱形水杯喝水,他每天喝( )杯水比较合适。
平顶山2024年05版小学六年级第四次英语第五单元综合卷

平顶山2024年05版小学六年级英语第五单元综合卷考试时间:90分钟(总分:110)考试人:_________题号一二三总分得分评级介绍:这份试卷涵盖小学六年级第五单元的核心知识,难度适中,重点关注词汇、语法和日常对话能力,旨在提升学生的英语综合运用能力。
一、(选择题)总分:20分(2分/题)1、What do you call the place where you watch movies?A, TheaterB, MuseumC, LibraryD, Gallery2、How many hours are in a day?A, 12B, 24C, 36D, 483、How do you say "家" in English?A, HomeB, SchoolC, Work4、What is the English translation of "食用植物"?A, Edible plantsB, Inedible plantsC, Poisonous plantsD, Ornamental plants5、What do we call the time when plants stop growing?我们称植物停止生长的时间为?A, DormancyB, Growth periodC, ReproductionD, Germination6、What is the English translation of "植物繁殖"? A, Plant reproductionB, Plant growthC, Plant cultivationD, Plant management7、Which word means "to give a speech"?A, SpeakB, TalkC, AnnounceD, Present8、What is the English word for "美洲豹"?A, CheetahB, LeopardC, JaguarD, Lynx9、What does "小鹿" mean in English?A, FawnB, CalfC, KidD, Cub10、How do you feel when you win a game?A, SadB, HappyC, AngryD, Confused11、What do you call someone who writes books? A, PainterB, AuthorC, ScientistD, Musician12、What is the main function of leaves?A, To absorb waterB, To produce foodC, To grow rootsD, To attract insects13、What do we call a plant that has a short life cycle?中文解释:我们称生命周期短的植物为?A, AnnualB, PerennialC, Biennial14、This person is a successful businessman. 这个人是一位成功的商人。
银川2024年05版小学四年级上册第4次英语第6单元综合卷

银川2024年05版小学四年级上册英语第6单元综合卷考试时间:80分钟(总分:110)A卷考试人:_________题号一二三四五总分得分一、综合题(共计100题)1、What do we call the frozen form of water?A. SteamB. IceC. LiquidD. Vapor答案:B2、听力题:The __________ can change due to erosion and deposition.3、What is the main purpose of a map?A. To tell timeB. To show directionsC. To measure distancesD. To display pictures答案: B4、填空题:My _____ (表妹) is very funny.5、听力题:The three states of matter are solid, liquid, and ______.6、填空题:I saw a ________ fluttering near the flowers.7、听力题:They are ___ ice cream. (eating)8、What is the term for a baby goat?A. CalfC. LambD. Fawn答案:B9、填空题:We enjoy ______ (看电视) in the evenings.10、Which color is a stop sign?A. BlueB. YellowC. RedD. Green答案:C11、听力题:A solution where no more solute can dissolve is called ______.12、听力题:The __________ is a famous area known for its outdoor activities.13、听力题:The _____ (ocean/lake) is calm.14、填空题:The ancient civilization of ________ is known for its philosophical teachings.15、填空题:The ________ was a major event in American history that led to independence.16、听力题:She is _____ (writing) in her notebook.17、填空题:The first president of the United States was __________ (乔治·华盛顿).18、What do we call the time when we eat in the morning?A. DinnerB. LunchC. BreakfastD. Snack答案:C19、What is the main ingredient in a scone?A. FlourB. SugarD. Egg答案:A20、填空题:A small ___ (小白兔) hops through the meadow.21、Which bird is known for its colorful feathers?A. PigeonB. SparrowC. PeacockD. Crow答案:C22、What is the main source of energy for the Earth?A. MoonB. SunC. StarsD. Wind答案:B23、填空题:The hedgehog rolls into a ball when ________________ (害怕).24、填空题:My cousin has a __________狗. (可爱的)25、Which animal is known for building dams?A. BeaverB. OtterC. RabbitD. Squirrel答案:A26、填空题:________ (植物繁殖) can happen naturally or artificially.27、听力题:Bees help ______ plants by pollinating.28、听力题:Metals are typically _____ conductors of electricity.29、听力题:Matter is anything that has ______.The _______ (马) pulls a cart.31、听力题:The capital of the Solomon Islands is __________.32、选择题:What is the opposite of sweet?A. SourB. BitterC. SpicyD. Salty33、听力题:I have a ___ (sister/brother) and a dog.34、填空题:The _____ (小花) bloomed beautifully in the garden.35、选择题:What is the opposite of inside?A. OutB. WithinC. CloseD. Near36、填空题:My dad loves __________. (烹饪)37、填空题:A _____ (植物成长) can lead to community beautification.38、听力题:The Earth's surface contains various ______ features.39、What is the capital of Fiji?A. SuvaB. LautokaC. NadiD. Labasa答案:A40、听力题:The _______ of an object can be tested using a force meter.Indicators help us determine whether a solution is _____ or basic.42、填空题:My dog enjoys rolling in the _______ (草).43、What is the name of the famous river that runs through Egypt?A. NileB. AmazonC. YangtzeD. Mississippi答案:A44、What do we call the act of using your senses to gather information?A. ObservationB. ExplanationC. PredictionD. Conclusion答案:A45、填空题:The __________ (历史的价值观) reflect societal beliefs.46、What is the capital of Kenya?A. NairobiB. KampalaC. Addis AbabaD. Dar es Salaam答案:A47、What do we call the long period of time before we enter adulthood?A. ChildhoodB. AdolescenceC. TeenagerD. Infancy答案:A. Childhood48、听力题:We will _______ (have) a barbecue this weekend.49、填空题:My favorite teacher is _______ (名字). 她教我们 _______ (科目).50、What do we call the process of heating food to make it safe to eat?A. FreezingB. CookingC. ChillingD. Baking答案:B51、填空题:A puppy needs a lot of ______ (爱).52、What is the main ingredient in a Caesar salad?A. LettuceB. SpinachC. KaleD. Arugula答案:A. Lettuce53、选择题:What do we call the action of looking at stars?A. StargazingB. SkywatchingC. ObservingD. Gazing54、What is the capital of Bolivia?A. SucreB. La PazC. Santa CruzD. Cochabamba55、听力题:A compass uses the Earth’s magnetic ______ to find direction.56、填空题:The _______ (The Harlem Renaissance) showcased African American culture and arts.57、Which one is a type of weather?A. RainB. ChairC. RoadD. Tree58、填空题:Many plants have adapted to survive in nutrient-poor ______.(许多植物已适应在养分贫乏的土壤中生存。
人教A版高三数学选修4-5模块综合检测卷

模块综合检测卷(测试时间:120分钟,评价分值:150分)一、选择题(每小题5分,共60分)1.用数学归纳法证明3n >n 3(n ≥3,n ∈N)第一步应验证( ) A .n =1 B .n =2 C .n =3 D .n =4 答案: C2.不等式|3x -2|<4的解集是( ) A.{}x |x >2 B.⎩⎨⎧⎭⎬⎫x ⎪⎪⎪x <-23C.⎩⎨⎧⎭⎬⎫x ⎪⎪⎪x <-23或x >2D.⎩⎨⎧⎭⎬⎫x ⎪⎪⎪-23<x <2 答案: D3.已知a ,b ,c ,d ∈R ,且ab >0,-ca <-d b,则下列各式恒成立的是( ) A .bc <ad B .bc >ad C.a c >b d D.a c <b d答案: B4.若a ,b ,x ,y ∈R ,则⎩⎪⎨⎪⎧x +y >a +b ,(x -a )(y -b )>0是⎩⎪⎨⎪⎧x >a ,y >b 成立的( ) A .充分而不必要条件 B .必要而不充分条件 C .充要条件D .既不充分也不必要条件 答案: C5.给出三个条件:①ac 2>bc 2;②a c >b c;③a 2>b 2.其中能分别成为a >b 的充分条件的个数为( ) A .0个 B .1个 C .2个 D .3个 答案: B6.若a >0,使不等式|x -4|+|x -3|<a 在R 上的解集不是空集的a 的取值范围是( ) A .0<a <1 B .a =1 C .a ≥1 D .a >1 答案: D7.设x >0,y >0且x +y ≤4,则下列不等式中恒成立的是( )A.1x +y ≤14B.1x +1y≥1 C.xy ≥2 D.1xy ≥14答案: D8.若k 棱柱有f (k )个对角面,则k +1棱柱有对角面的个数为( ) A .2f (k ) B .k -1+f (k ) C .f (k )+k D .f (k )+2 答案: B9.已知f (x )是定义在正整数集上的函数,且f (x )满足:“当f (k )≥k 2成立时,总可推出f (k +1)≥(k +1)2成立”,那么下列命题总成立的是( )A .若f (3)≥9成立,则当k ≥1时,均有f (k )≥k 2成立B .若f (4)≥16成立,则当k ≥4时,均有f (k )<k 2成立C .若f (7)≥49成立,则当k <7时,均有f (k )<k 2成立D .若f (4)=25成立,则当k ≥4时,均有f (k )≥k 2成立解析:∵f (k )≥k 2成立时f (k +1)≥(k +1)2成立.当k =4时,f (4)=25>16=42成立,∴当k ≥4时,有f (k )≥k 2恒成立.答案:D10.用数学归纳法证明对一切大于1的自然数n ,不等式⎝ ⎛⎭⎪⎫1+13⎝ ⎛⎭⎪⎫1+15·…·⎝ ⎛⎭⎪⎫1+12n -1>2n +12成立,当n =2时验证的不等式是( )A .1+13>52B.⎝ ⎛⎭⎪⎫1+13⎝ ⎛⎭⎪⎫1+15>52C.⎝ ⎛⎭⎪⎫1+13⎝ ⎛⎭⎪⎫1+15≥52D .以上都不对解析:当n =2时,左边=1+12×2-1=1+13,右边=2×2+12=52,∴应验证1+13>52.答案:A11.用数学归纳法证明“对于任意x >0时的正整数n ,都有x n +x n -2+x n -4+…+1x n -4+1x n -2+1xn ≥n+1”时,需验证的使命题成立的最小正整数值n 0应为( )A .n 0=1B .n 0=2C .n 0=1,2D .以上答案均不正确解析:∵n ∈N +,∴n 的最小值为1,即n 0=1. 答案:A 12.记满足下列条件的函数f (x )的集合为M ,当|x 1|≤1,|x 2|≤1时,|f (x 1)-f (x 2)|≤4|x 1-x 2|,又令g (x )=x 2+2x -1(|x |≤1),则g (x )与M 的关系是( )A .g (x )MB .g (x )∈MC .g (x )∉MD .不能确定解析:因为g (x 1)-g (x 2)=x 21+2x 1-x 22-2x 2=(x 1-x 2)(x 1+x 2+2),|g (x 1)-g (x 2)|=|x 1-x 2|·|x 1+x 2+2|≤|x 1-x 2|(|x 1|+|x 2|+2)≤4|x 1-x 2|,所以g (x )∈M .答案:B二、填空题(每小题5分,共20分)13.函数y =3x +4x2(x >0)的最小值为________.答案:33914.x ,y ∈R ,若x +y =1,则x 2+y 2的最小值为________.答案:1215.设数列{a n }满足a 1=2,a n +1=2a n +2,用数学归纳法证明a n =4×2n -1-2的第二步中,设n=k 时结论成立,即a k =4×2k -1-2,那么当n =k +1时,______________________________________.答案:a k +1=2a k +2=2(4×2k -1-2)+2=4×2k -2=4×2(k +1)-1-216.不等式|x +3|-|x -1|≤a 2-3a 对任意实数x 恒成立,则实数a 的取值范围为________. 答案:(-∞,-1]∪[4,+∞)三、解答题(本大题共6小题,共70分)17.(本小题满分11分)已知a 、b 、c ∈R +,求证:b +c -a a +c +a -b b +b +a -c c≥3.证明:∵a 、b 、c ∈R +,b +c -a a +c +a -b b +b +a -c c =b a +c a -1+c b +a b -1+b c +a c -1=⎝ ⎛⎭⎪⎫b a +c b +a c +⎝ ⎛⎭⎪⎫c a +b c +a b -3≥33b a ×c b ×a c +33c a ×b c ×a b -3=3.当且仅当a =b =c 时等号成立.18.(本小题满分11分)已知关于x 的不等式|ax -1|+|ax -a |≥1(a >0). (1)当a =1时,求此不等式的解集;(2)若不等式的解集为R ,求实数a 的取值范围. 解析:(1)当a =1时,得2|x -1|≥1.∴x ≥32或x ≤12.∴不等式的解集为⎝ ⎛⎦⎥⎤-∞,12∪⎣⎢⎡⎭⎪⎫32,+∞. (2)∵原不等式的解集为R ,∴|ax -1|+|ax -a |≥1对一切实数x 恒成立. 又∵|ax -1|+|ax -a |≥|a -1|, ∴|a -1|≥1, ∴a ≥2或a ≤0. ∵a >0,∴a 的取值范围为[2,+∞).19.(本小题满分12分)设x >0,y >0,证明:(x 2+y 2)12>(x 3+y 3)13.证明:证法一(分析法) 所证不等式等价于(x 2+y 2)3>(x 3+y 3)2,即x 6+y 6+3x 2y 2(x 2+y 2)>x 6+y 6+2x 3y 3,即3x 2y 2(x 2+y 2)>2x 3y 3,只需证:x 2+y 2>23xy ,∵x 2+y 2≥2xy >23xy 成立,∴(x 2+y 2)12>(x 3+y 3)13,证法二(综合法) ∵(x 2+y 2)3=x 6+y 6+3x 2y 2(x 2+y 2)≥x 6+y 6+6x 3y 3>x 6+y 6+2x 3y 3=(x 3+y 3)2,∵x >0,y >0,∴(x 2+y 2)12>(x 3+y 3)13.20.(本小题满分12分)已知a >b >c >0,方程x 2-(a +b +c )x +ab +bc +ca =0,若该方程有实根,求证:a ,b ,c 不能成为一个三角形的三边长.证明:∵方程x 2-(a +b +c )x +ab +bc +ca =0有实根,∴Δ=(a +b +c )2-4(ab +bc +ca ) =a 2+b 2+c 2-2(ab +bc +ca ) =(a -b )2-2(a +b )c +c 2=[(a +b )2-c ]·[(a -b )2-c ] =(a +b +c )(a +b -c )(a -b +c )(a -b -c )≥0.若a ,b ,c 为一个三角形的三边长,由a +b +c >0, a +b -c >0,a -b +c >0得a -b -c ≥0,即b +c ≤a ,即b +c <a .这与三角形两边之和大于第三边矛盾. ∴a ,b ,c 不能成为一个三角形的三边长.21.(本小题满分12分)已知函数f (x )=x +3x +1(x ≠-1),设数列{a n }满足a 1=1,a n +1=f (a n ),数列{b n }满足b n =|a n -3|,S n =b 1+b 2+…+b n (n ∈N *).(1)用数学归纳法证明:b n ≤(3-1)n2n -1; (2)求证:S n <233.证明:(1)当x ≥0时,f (x )=1+2x +1≥1, 因为a 1=1,所以a n ≥1(n ∈N *),下面用数学归纳法证明不等式b n ≤(3-1)n2n -1: ①当n =1时,b 1=3-1,不等式成立.②假设当n =k (k ≥1,k ∈N *)时,不等式成立,即b k ≤(3-1)k2k -1, 那么n =k +1时,b k +1=|a k +1-3|=(3-1)|a k -3|1+a k ≤3-12b k ≤(3-1)k +12k. 所以,当n =k +1时,不等式也成立.由①②可知不等式对任意n ∈N *都成立.(2)由(1)知b n ≤(3-1)n2n -1, 所以S n =b 1+b 2+…+b n ≤(3-1)+(3-1)22+…+(3-1)n2n -1=(3-1)×1-⎝ ⎛⎭⎪⎫3-12n 1-3-12<(3-1)×11-3-12=233.故对任意n ∈N *,S n <233.22.(本小题满分12分)已知数列{b n }是等差数列,且b 1=1,b 1+b 2+…+b 10=145(n ∈N *). (1)求数列{b n }的通项;(2)设数列{a n }的通项a n =log a ⎝⎛⎭⎪⎫1+1bn(其中a >0且a ≠1),设S n 是数列{a n }的前n 项和,试比较S n 与13log a b n +1的大小,并证明你的结论.解析:(1)设数列{b n }的公差为d ,由题意,得10×1+10×(10-1)2×d =145,∴d =3,b n =3n-2.(2)由b n =3n -2,知S n =log a (1+1)+log a ⎝⎛⎭⎪⎫1+14+…+log a (1+13n -2) =log a ⎣⎢⎡⎦⎥⎤(1+1)⎝ ⎛⎭⎪⎫1+14…⎝ ⎛⎭⎪⎫1+13n -2,13log a b n +1=log a 33n +1, 因此要比较S n 与13log a b n +1的大小,可先比较(1+1)·⎝ ⎛⎭⎪⎫1+14…⎝ ⎛⎭⎪⎫1+13n -2与33n +1的大小,取n =1,有(1+1)>33×1+1,猜想取n ≥1,n ∈N *,有(1+1)⎝ ⎛⎭⎪⎫1+14…(1+13n -2)>33n +1, 下面用数学归纳法证明之:①当n =1时,已验证不等式成立.②假设当n =k (k ∈N *)时不等式成立,即(1+1)⎝ ⎛⎭⎪⎫1+14…⎝ ⎛⎭⎪⎫1+13k -2>33k +1,则当n =k +1时,(1+1)⎝ ⎛⎭⎪⎫1+14…⎝ ⎛⎭⎪⎫1+13k -2⎣⎢⎡⎦⎥⎤1+13(k +1)-2> 33k +1⎝ ⎛⎭⎪⎫1+13k +1=33k +13k +1·(3k +2), ∵⎣⎢⎢⎡⎦⎥⎥⎤33k +13k +1(3k +2)3-(33k +4)3= (3k +2)3-(3k +4)(3k +1)2(3k +1)2=9k +4(3k +1)2>0.∴33k +13k +1·(3k +2)>33k +4=33(k +1)+1, 因此(1+1)(1+14)…(1+13k -2)[1+13(k +1)-2]>33(k +1)+1.这说明,当n =k +1时不等式也成立.由①②知,对一切n ∈N *,不等式(1+1)(1+14)…⎝ ⎛⎭⎪⎫1+13n -2>33n -1都成立.再由对数性质,可得:当a >1时,S n >13log a b n +1;当0<a <1时,S n <13log a b n +1.。
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Exam for Module 6bridge. I'd come to this comer and studied the bridge a million times. And on every one of those times, I was taken with what I'd come to call Brooklyn Belle. I never got tired of looking out at Belle. Now she was a powerful giant who carried all kinds of people to all kinds of places, day after day.At night Belle was dressed in tiny lights that spanned her limbs. On a cloudless night like this one, she was a sight like no other sight in the whole city. Jeweled in light. Beautiful.My fingers had tensed into fists at my sides, fists full of strength and eagerness. Then I reached into my jacket pocket and pulled out my pencil. Slowly, I flipped through the pages of my sketchbook. I'd drawn Belle in the high-noon light, at sunset, on snowy days, and on foggy mornings. My favorite sketches were those of Belle during rush hour, when cars and taxis danced along her outstretched beams.Tonight I'd draw Belle with her lighted cape. I sketched slowly at first, then faster, my pencil working with the speed of my excitement the thrill that worked me over every time I sketched that bridge.I was proud of my drawings(I liked to think of them as portraits ),but with each page they showed a sad truth about Belle: She needed repair. .She was some forty years older than my grandmother. And as lovely as she was, she had some serious rough spots-corroded cables, rust. and chipped paint that had built up over the decades. That bridge renovation project needed me; and I needed it, in more ways than I could count.21.When does this passage take place?A. At night.B .At sunset.C. On a snowy day.D. On a foggy morning.22.The author refers to the Brooklyn Bridge as"Brooklyn Belle” to help the reader__________.A. understand that the bridge is very oldB. remember the bridge's historyC. imagine the noise around the bridgeD. appreciate the bridge's beauty23.In the passage, the narrator refers to the bridge renovation project. Based on information in the passage, the “renovation” project most likely refers to _____________.A repairing the bridgeB .removing the bridgeC. replacing the bridgeD. remembering the bridgeBIn Antarctica, you learn not to take anything for granted. Everything people depend on hasto be shipped or flown into the continent. Why? There are no farms on the icy continent. There are certainly no cows, pigs or chickens. As for drinking water, special systems and a great deal of energy are needed to take the salt out of seawater to make it useable.Then there is the matter of waste. The US Antarctic Program is committed(致力于)to reducing its impact-or footprint-on the Antarctic environment. That means that every bit of garbage a person might produce in a day has to be transported off the continent That's true whether it's the wrapper from your candy bar, the green beans you didn't want to eat or the paper towels you used to wipe your hands.Mcmurdo Station has put into place technology that helps it reduce its footprint. Itstarts with energy. The US research base has partnered its nearby neighbor, New Zealand's Scott Base, to share energy produced from three large wind turbines. Scott Base gets 100% of its power from wind, says Ron Blevins, manager at a power plant, while the American base gets about 359% of its energy from wind. The much larger US base uses oil-fired generators(发电机)to supply the rest of its energy needs. The waste heat produced by the generators is then used to warm many of its buildings.The Mcmurdo base also has a wastewater treatment plant. Yubecca Bragg, an organic farmer, manages the treatment plant. Bragg explains that treatment depends on allowing microorganisms(微生物)to break down the waste until the liquid part of the waste can be safely released into the ocean. Between 150,000 and 180,000 pounds a year of the remaining solid waste is packed into containers and sent back to the US.24.We can learn from the first paragraph that____________.A. it is not easy to live in AntarcticaB. water in Antarctica is useless at allC. farmers have to raise cows in AntarcticaD. people like taking the ships in Antarctica25.How do people in Antarctica deal with the solid garbage?A. They ignore it completely..B. They keep it in the dustbins.C. They have it taken away later.D. They wrap it in the paper towels.26.It can be inferred from Ron Blevins' words that_____________.A. Scott Base tries to remove its footprintB. New Zealand's technology is advancedC. the US base gets its most energy from windD. the US base doesn't need too much energy27.Which of the following is harmful to the environment of Antarctica?A.A wastewater treatment plant.B. Scott base's power plant.C. The large wind turbine.D .An oil-fired generator.CMy mother and I came from Hungary to America in the early 1950s.I cherish the fun we have working together to adapt to our new life in America.In the morning Mother and I travel together to work. Rising at dawn, I love the adventureof tiptoeing around in the dark living room where we sleep, whispering to each other so as not to wake Aunt Celia and Uncle Martin in the bedroom, then leaving the house soundlessly and making our way to the subway station in semidarkness There is a sense of mystery -a sense of power- in being here before the rest of the world awakens, as if witnessing the beginning of time.We love travelling together on the subway. It's fun to observe our fellow subway riders, exchange jokes in Hungarian, play guessing games as to their identities, their ages, and their jobs and make bets as to where they'd be getting off.In a couple of weeks Mother become. familiar with the train route, and she no longer allows me to accompany her.“But I'll miss the fun of raveling together nthe morning," I protest.“So will 1. But in order to accompany me. you must rise an hour earlier, losing an hours sleep."Mother is firm, and I have no choice but to obey and regretfully give up on our morning fun."I hope you'll have time to teach me English I don't want to depend on you every time I need to make a phone call in English, or want to take public transport.”“Okay, madam,”I agree in a happy tone. "How about today? Let's have our first lesson this evening!”Every evening I grill Mother in vocabulary and grammar, and indeed in a couple of weeks she learns enough English to do marketing on her own and travel freely by subway.28.According to information in the passage, it is most important to the mother that she______.A. tell jokes in HungarianB. rise early in the morningC. learn to speak in EnglishD.travel daily on the subway29.Which detail from the passage best shows that the writer is excited about her life in America?A. Her resistance to the growing independence of her mother.B. Her acceptance of why she can no longer accompany her mother.C. Her discipline to work on language lessons every day with her mother.D. Her description of rising early to ride the subway with her mother.30.Which word best describes the writers behavior toward her mother?A. Disrespectful.B. Angry.C. Supportive.D. Sympathetic.Ⅲ.Complete the following passage with appropriate sentences. Two are extra.Easy Ways to Get HealthyI always hate the first few weeks of the new year ,mostly because I'm sick to death of people in commercials who are all saying, "Let's diet! It's a new year!”You can always choose to switch it up! 31.________________ Here are some simple ways to get healthy on all year around.Switch out bad habits for good ones. 32.______________ Do something as simple as swapping nighttime cookies for nighttime vitamins.Catch some sleep. One of the most important things we can do for our minds and bodies isto make sure we're getting enough sleep on a regular schedule.33.______________ Feed your mind. Keep your mind flexible and smart. Do crossroads, Sudoku, start a new hobby or read a book about something you don't understand-anything that challenges you to think differently and exercise the brain of yours.34.________________ I'm not suggesting you take on a brand new fitness quest, but take the stairs, park further away, and find a hobby that allows you to be active.Get organized. I'm not saying a whole repair of your entire world; start small. Organize a kitchen cabinet. Purge yourself of stuff you don't need.Get rid of the mess. That means your mind is included too! Make lists. Get a calendar and mark down important days. Clear your mind and your home. 35._______________ Smile. Overlook some of the bad things that are going on in your life, and focus on the good. There are positives in nearly every situation, even if it doesn't feel like it. Staying positive can helpⅣ.Cloze test.I was watching this movie on television the other day about a man who did whatever it took to get money. To him money was 36_____.He killed his own brother and even his wife to 37______ what he wanted. This man didn't allow anything or anyone to 38________ him.Money is a resource of power and status. People who 39_____ a lot of money also have a lot of money to spend. We have always considered 40______to be powerful people. A lot of money allows people to 41________ other people as well.Do parents realize that kids at a(n)42_________ age recognize that having money to spend provides them 43__________ a feeling of power? It 44___________ them to take control of a status. Teens learn this feeling of power from adults 45______ they observe parents showing off their purchases or 46_____ to neighbors about their new BMW. Teens quickly link 47____ with success and power.Who are the most powerful people in the world? Bill Gates? President George Bush? Takea look at the well-respected 48____magazine, Forbes. Each year Forbes 49____a list of the Top CEOs-Corporate Most Powerful People . The list is based on how much money each CEO 50______during the preceding year.But do you know that not all powerful people are 51.______? Some have been very poor, like our dear Mother Theresa, the nun(修女) who worked tirelessly to 52____ the world's poorest and most desperate people. She could do all this 53.____ she had courage and belief and most 54._____ from the love of her heart.Is money powerful? What do we want to tell our teens? Let's give much 55.____ to this.36.A.time B. power C. knowledge D. wealth37.A.have B.keep C.get D.give38.A.stop B.help C.pause D.quit39.A.collect B.make C.spend D.waste40.A.her B.him C.it D.them41.A.feed B.fool C.control D.respect42.A.small B.old C.early te43.A.for B.with C.to D.by44.A.lets B.has C.promises D.allows45.A.what B.that C.since D.when46.A.listening B.talking C.replying D.turning47.A.power B.purchases C.money D.cars48.A.business B.tourism eation D.fashion49.A.shows B.prints C.releases D.reports50.A.eamed B.spent C.cost D.lost51.A.poor B.lucky C.powerful D.rich52.A.care for B.look for C.call for D.long for53.A.while B.before C.as D.after54.A.proudly B.likely C.importantly D.possibly55.A.care B.thought C.idea D.attitudeV. Fill in each blank with a proper word ,or with the proper form of the word given in the brackets.Air pollution is no doubt a major concern in many big cities across the world. More than 80% of people living in urban areas 56.______ (expose)to unsafe air quality, according to a report 57.________ (deliver)by the World Health Organization Thursday.“Urban air pollution continues to rise at alarming rate, doing great harm to58._______ human health, "WHO Director of Public Health and Environmental Policy Dr Maria Neira said in the report.“At the same time,59.________ (aware)is rising and more cities are monitoring their air quality.”In the past two years, the number of cities monitoring air pollution 60._________ (double) now-covering 3,000 cities in 103 countries.The WHO warned people of the serious effects poor air quality could have 61.________ the health of urban residents,62.________ (link) it to risk of stroke, heart disease, lung cancer and many other diseases.The report also showed that low-and-middle-income countries generally have 63._______ (poor)air quality. About 98% of cities with 100,000 or more people in low-and-middle-income countries do not meet WHO air quality guidelines,64.__________ 56% in high-income countries fail to meet the standard. The report also urged local governments to take stronger and more 65._______ (effect)measures to fight against air pollution.VI. Practical writing.假设你叫张华,作为学校国际交流团成员,刚刚结束了对美国的友好学校的访问。